1209.2019 (Phil Broadbridge et al.)

Solutions of Helmholtz and Schrödinger equations with side condition
and nonregular separation of variables    [PDF]

Phil Broadbridge, Claudia M. Chanu, W. Miller Jr

Olver and Rosenau (1986) studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply their idea to the special case of finite dimensional Hamiltonian systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schr\"odinger equations with potential on N-dimensional Riemannian and pseudo-Riemannian manifolds, where more structure is available. We show that the requirement of N-1 commuting 2nd order symmetry operators, modulo a 2nd order side condition corresponds to nonregular separation of variables in an orthogonal coordinate system, characterized by a generalized St\"ackel matrix. The coordinates and solutions obtainable through true nonregular separation are distinct from those arising through regular separation of variables. We develop the theory for these systems and provide examples

View original: http://arxiv.org/abs/1209.2019